The Jacobi operator and its Donoghue $m$-functions

In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, associated to the differential expression \begin{align*} \begin{split} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha + 1}(1+x)^{\beta + 1}\big) (d/dx),& \\ x \in (-1,1), \; \alpha, \beta \in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's $m$-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature.